Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 2nd Term
Week: 4
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Vectors in Two Dimensions I.
Topic: FUNCTIONS
Subject Matter: logarithmic function, exponential function, application of functions, solving problems involving functions

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define logarithmic and exponential functions.
Identify the characteristics of logarithmic and exponential graphs.
State real-life applications of functions.
Outline steps for solving problems involving functions.

Affective Domain:

Develop interest in applying functions to real-world scenarios.
Appreciate the importance of functions in solving mathematical problems.
Work cooperatively with peers to solve functional problems.

Psychomotor Domain:

Sketch simple graphs of logarithmic and exponential functions.
Apply the properties of logarithmic and exponential functions to solve problems.
Accurately solve problems involving various types of functions.

Participate actively in class discussions about functions.
Collaborate with classmates during problem-solving activities.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Senior Secondary Schools.
State Unified Scheme of Work for Further Mathematics SSS 1.
Further Mathematics for Senior Secondary Schools 1 by P.N. Anaekwe et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

Whiteboard/Chalkboard
Markers/Chalk
Charts of solved function problems
Further Mathematics textbooks

Rationale for the Lesson

This lesson helps pupils understand how different types of functions work and how they are used to model real-life situations. This understanding is important for solving various problems in science, engineering, and economics, providing a foundation for advanced mathematical concepts.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic algebra, concept of functions, inverse operations, and graphical representation of simple functions like linear and quadratic equations.

Lesson Content/Board Summary

FUNCTIONS

1. Logarithmic Function

A logarithmic function is the inverse of an exponential function. It is written as y = log_bx, where b is the base (b > 0, b ≠ 1) and x > 0.

The following are key properties of logarithmic functions:

log_b(MN) = log_bM + log_bN
log_b(M/N) = log_bM – log_bN
log_b(M^p) = p log_bM
log_bb = 1
log_b1 = 0

2. Exponential Function

An exponential function is a mathematical function of the form f(x) = a^x, where a is a constant called the base (a > 0, a ≠ 1) and x is the exponent or independent variable.

The following are characteristics of exponential functions:

The domain is all real numbers.
The range is all positive real numbers (if a > 0).
The graph passes through (0, 1).
If a > 1, the function is increasing.
If 0 < a < 1, the function is decreasing.

3. Application of Functions

Functions are used to model various real-life situations, showing relationships between quantities.

The following are areas where functions are applied:

Population growth and decay (exponential functions).
Compound interest calculations (exponential functions).
Sound intensity measurement (logarithmic functions).
Radioactive decay (exponential functions).
Modelling costs, revenues, and profits in business.

4. Solving Problems Involving Functions

Solving problems involving functions often requires understanding the type of function and applying its properties or rules.

The following are general steps for solving functional problems:

Understand the problem and identify the given function or relationship.
Determine the type of function (linear, quadratic, exponential, logarithmic, etc.).
Apply relevant formulas, properties, or theorems of that function type.
Substitute given values into the function or equation.
Perform calculations carefully to find the unknown.
Interpret the result in the context of the problem.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher introduces the lesson by reviewing pupils’ prior knowledge of basic functions and their graphs. The teacher asks pupils to give examples of relationships that can be represented mathematically.
Pupils’ Activity: Pupils respond to questions and share examples of relationships they know.
Learning Point: Pupils recall basic concepts of functions and their representation.

Step 2: Logarithmic Functions

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines logarithmic functions, explains their relationship with exponential functions, and discusses their basic properties. The teacher uses charts to show typical graphs of logarithmic functions.
Pupils’ Activity: Pupils listen attentively, take notes, and ask questions for clarification.
Learning Point: Pupils understand the definition and basic properties of logarithmic functions.

Step 3: Exponential Functions

Time: 8 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher defines exponential functions, explains their characteristics, and illustrates their graphs using charts. The teacher provides simple examples to demonstrate their behaviour.
Pupils’ Activity: Pupils observe the charts, listen to explanations, and copy examples into their notebooks.
Learning Point: Pupils understand the definition and characteristics of exponential functions.

Step 4: Application of Functions

Time: 7 minutes
Teaching Skill: Real-world Connection
Teacher’s Activity: The teacher discusses various real-life applications of functions, particularly logarithmic and exponential functions, giving examples from population growth, finance, and science. The teacher encourages pupils to think of other applications.
Pupils’ Activity: Pupils listen, contribute ideas, and note down the applications discussed.
Learning Point: Pupils recognize the relevance of functions in practical situations.

Step 5: Solving Problems Involving Functions

Time: 7 minutes
Teaching Skill: Problem Solving/Drilling
Teacher’s Activity: The teacher explains the general steps for solving problems involving different types of functions. The teacher demonstrates solving a few problems from the charts, guiding pupils through each step.
Pupils’ Activity: Pupils observe the steps, ask questions, and follow the teacher’s examples.
Learning Point: Pupils learn a systematic approach to solving functional problems.

Step 6: Class Activity/Practice

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides a simple problem involving either a logarithmic or exponential function for pupils to solve individually or in pairs. The teacher moves around to assist pupils.
Pupils’ Activity: Pupils attempt to solve the given problem, applying the steps learned.
Learning Point: Pupils practice applying their understanding to solve functional problems.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define a logarithmic function.
State two properties of an exponential function.
Mention two real-life applications of functions.
Outline the general steps for solving problems involving functions.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the main points of the lesson, reiterating the definitions, properties, applications, and problem-solving steps for functions. The teacher assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their learning and prepare for independent practice.

Lesson Keywords

Function – A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Logarithmic Function – The inverse of an exponential function, expressed as y = log_bx.
Exponential Function – A function of the form f(x) = a^x, where the base ‘a’ is a positive constant and not equal to 1.
Base – The number that is raised to a power in an exponential expression, or the number whose logarithm is taken.
Domain – The set of all possible input values (x-values) for which a function is defined.
Range – The set of all possible output values (y-values) that a function can produce.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems or challenge them to research additional applications of functions. For pupils needing more support, the teacher can offer simplified examples, provide one-on-one guidance, or allow them to work in small groups with peer support.

Note for teachers using this lesson plan

Teachers should ensure that pupils have a solid grasp of basic algebraic operations before introducing complex functional problems. Encourage visual aids and real-world examples to make the concepts more relatable. Allocate sufficient time for practical exercises to reinforce understanding.